On an extension of a theorem on conjugacy class sizes.
The authors generalise a result of the reviewer in 1972 [J. Lond. Math. Soc., II. Ser. 5, 127-132 (1972; Zbl 0242.20025)] in which it was proved that if the conjugacy classes of a finite group have sizes precisely \(\{1,p^a,q^b,p^aq^b\}\) where \(p^a\) and \(q^b\) are prime powers then the group is nilpotent. Later \textit{A. Beltrán} and \textit{M. J. Felipe} generalised this in a series of papers [J. Group Theory 9, No. 6, 787-797 (2006; Zbl 1108.20021), J. Algebra 296, No. 1, 253-266 (2006; Zbl 1091.20017), ibid. 320, No. 12, 4317-4319 (2008; Zbl 1155.20306)] to prove that if the conjugacy classes of a finite group have sizes precisely \(\{1,m,n,mn\}\) where \(m\) and \(n\) are coprime then \(G\) is nilpotent. The authors consider what happens when there are only conditions are biprimary elements in a finite group. An element \(g\) of a group is called `biprimary' if its order is divisible by at most two primes. So the main result in this paper is: If the conjugacy class sizes of biprimary elements of a finite group have sizes precisely \(\{1,p^a,m,p^am\}\) where \(p^a\) is a prime power and \(m\) is prime to \(p\) and further there is a \(p\)-element whose conjugacy class size is \(p^a\) then \(G\) is nilpotent.
scientific article; zbMATH DE number 5823091